Essential Spectrum of Discrete Laplacian – Revisited

Abstract

Consider the discrete Laplacian operator A acting on l2(Z). It is well known from the classical literature that the essential spectrum of A is a compact interval. In this article, we give an elementary proof for this result, using the finite-dimensional truncations An of A. We do not rely on symbol analysis or any infinite-dimensional arguments. Instead, we consider the eigenvalue-sequences of the truncations An and make use of the filtration techniques due to Arveson. Usage of such techniques to the discrete Schrödinger operator and to the multi-dimensional settings will be interesting future problems.

Integrated Stochastic Investigation of Singularly Perturbed Delay Differential Equations for the Neuronal Variability Model

“…Te solution of SPDDEs difers rapidly in the region which is known as layers that may be obvious in the solution or its gradient and frequently seem at the boundary region. Several problems in science as well as in engineering, elasticity, control theory, biosciences, and fuid mechanics are created by SPDDEs, such as those present in red cell models [16][17][18][19][20].…”